When the space \(M\)  is finite, the theorem above is proved in an elementary and elegant way (see Den Hollander [1]}, and Dembo and Zeitouni [2]}). In this work, we prove the theorem for general Polish metric spaces by extending this elementary proof via sequences of discretizations of the space.
We split the set \(M\)  in a finite number of subsets which belong to one of two distinct categories. The well-behaved sets are the ones with small diameter, while the badly behaved sets will have small \(\mu \) -measure. We remark though that when the space \(M\)  is compact, no badly behaved sets are necessary. These partitions define natural projections on the space and allow us to approximate the sequence \(\big (X_{n} \big )_{n \in \mathbb {N}}\)  by variables in the discretized spaces, and by consequence provide approximations for its empirical measures. The main technical observation is that the discretized relative entropy converges to the relative entropy (REF ) as we take thinner partitions (see Lemma ) and the relative entropy is well approximated in balls (see Lemma ).
